Assume one is given a polyhedral complex $P$ in $\mathbb{R}^n$. Now consider picking uniformly at random a $D \subseteq \{0,1\}^n$. Is there way to upper bound the probability that $D$ (a subset of the Boolean hypercube) is not a subset of any of the cells of $P$? (feel free to assume any parameters of $P$ that you think you need)
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